Subgroups — C₈₉×#29202 #36407050

Order 22784.

658 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2647	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2327	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2068	C₂	1	2	yes	maximal
4	C₈₉×#2066	C₂	1	2	yes	maximal
5	C₈₉×#1960	C₂	1	1	yes	maximal, characteristic
6	C₈₉×#1956	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1865	C₂	1	2	yes	maximal
8	C₈₉×#1863	C₂	1	2	yes	maximal
9	C₈₉×#1846	C₂	1	1	yes	maximal, characteristic
10	C₈₉×#1836	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#1768	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
14	C₂×#107×C₈₉	C₂²	1	2	yes	—
15	C₂×#106×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×#103×C₈₉	C₂²	1	2	yes	—
19	C₂×#103×C₈₉	C₂²	1	2	yes	—
20	C₂×#103×C₈₉	C₂²	1	2	yes	—
21	C₂×#103×C₈₉	C₂²	1	2	yes	—
22	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	#400×C₈₉	—	2	2	no	—
25	#398×C₈₉	—	2	2	no	—
26	#391×C₈₉	C₂²	1	2	yes	—
27	#389×C₈₉	C₂²	1	2	yes	—
28	#389×C₈₉	C₂²	1	2	yes	—
29	#387×C₈₉	—	2	2	no	—
30	#387×C₈₉	—	2	2	no	—
31	#384×C₈₉	C₂²	1	1	yes	characteristic
32	#382×C₈₉	C₂²	1	1	yes	characteristic
33	#382×C₈₉	C₂²	1	1	yes	characteristic
34	#376×C₈₉	C₂²	1	1	yes	characteristic
35	#340×C₈₉	C₂²	1	2	yes	—
36	#340×C₈₉	C₂²	1	2	yes	—
37	#336×C₈₉	C₂²	1	1	yes	characteristic
38	#336×C₈₉	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
40	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
44	C₄⋊Q₈×C₈₉	—	2	2	no	—
45	C₄⋊Q₈×C₈₉	Q₈	1	4	yes	—
46	C₄×Q₈×C₈₉	—	4	2	no	—
47	C₄×Q₈×C₈₉	—	4	2	no	—
48	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
49	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
50	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
51	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
52	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
53	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
56	C₂×C₄²×C₈₉	—	2	2	no	—
57	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
60	#107×C₈₉	—	4	2	no	—
61	#106×C₈₉	—	4	2	no	—
62	C₄⋊C₈×C₈₉	—	2	2	no	—
63	C₄⋊C₈×C₈₉	—	2	2	no	—
64	#103×C₈₉	—	2	4	no	—
65	#103×C₈₉	—	2	4	no	—
66	#103×C₈₉	—	4	2	no	—
67	#103×C₈₉	—	4	2	no	—
68	#97×C₈₉	—	2	2	no	—
69	C₄×C₈×C₈₉	—	2	2	no	—
70	#95×C₈₉	—	2	2	no	—
71	#95×C₈₉	—	2	2	no	—
72	#95×C₈₉	—	2	2	no	—
73	#95×C₈₉	—	2	2	no	—
74	#95×C₈₉	—	2	2	no	—
75	#95×C₈₉	—	2	2	no	—
76	#95×C₈₉	D₄	1	2	yes	—
77	#95×C₈₉	D₄	1	2	yes	—
78	C₂×Q₈×C₈₉	—	2	2	no	—
79	C₂×Q₈×C₈₉	—	2	2	no	—
80	C₂×Q₈×C₈₉	—	2	2	no	—
81	C₂×Q₈×C₈₉	—	2	2	no	—
82	C₂×Q₈×C₈₉	—	2	2	no	—
83	C₂×Q₈×C₈₉	C₂×Q₈	1	4	yes	—
84	C₂²×C₄×C₈₉	—	2	2	no	—
85	C₂²×C₄×C₈₉	—	2	2	no	—
86	C₂²×C₄×C₈₉	—	2	2	no	—
87	C₂²×C₄×C₈₉	—	2	2	no	—
88	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
93	C₂×C₈×C₈₉	—	2	1	no	—
94	C₂×C₈×C₈₉	—	2	1	no	—
95	C₂×C₈×C₈₉	—	2	1	no	—
96	C₂×C₈×C₈₉	—	2	1	no	—
97	C₂×C₈×C₈₉	—	2	1	no	—
98	C₂×C₈×C₈₉	—	2	1	no	—
99	C₂×C₈×C₈₉	—	2	1	no	—
100	C₂×C₈×C₈₉	—	2	1	no	—
101	C₂×C₈×C₈₉	—	2	1	no	—
102	C₂×C₈×C₈₉	—	2	1	no	—
103	C₂×C₈×C₈₉	#41	1	2	yes	—
104	C₂×C₈×C₈₉	#41	1	2	yes	—
105	C₄⋊C₄×C₈₉	—	2	8	no	—
106	C₄⋊C₄×C₈₉	—	2	2	no	—
107	C₄⋊C₄×C₈₉	—	2	2	no	—
108	C₄⋊C₄×C₈₉	—	2	2	no	—
109	C₄⋊C₄×C₈₉	—	2	2	no	—
110	C₄⋊C₄×C₈₉	—	2	2	no	—
111	C₄⋊C₄×C₈₉	—	2	2	no	—
112	C₄⋊C₄×C₈₉	—	4	4	no	—
113	C₄⋊C₄×C₈₉	—	8	2	no	—
114	C₄⋊C₄×C₈₉	—	8	2	no	—
115	C₄⋊C₄×C₈₉	—	8	2	no	—
116	C₄²×C₈₉	—	2	1	no	—
117	C₄²×C₈₉	—	2	1	no	—
118	C₄²×C₈₉	—	2	1	no	—
119	C₄²×C₈₉	—	2	1	no	—
120	C₄²×C₈₉	—	4	4	no	—
121	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
122	C₄²×C₈₉	#41	1	2	yes	—
123	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
124	Q₈×C₈₉	—	2	4	no	—
125	Q₈×C₈₉	—	2	4	no	—
126	Q₈×C₈₉	—	4	2	no	—
127	Q₈×C₈₉	—	4	2	no	—
128	C₂×C₄×C₈₉	—	2	2	no	—
129	C₂×C₄×C₈₉	—	2	2	no	—
130	C₂×C₄×C₈₉	—	2	4	no	—
131	C₂×C₄×C₈₉	—	2	1	no	—
132	C₂×C₄×C₈₉	—	2	1	no	—
133	C₂×C₄×C₈₉	—	2	2	no	—
134	C₂×C₄×C₈₉	—	2	2	no	—
135	C₂×C₄×C₈₉	—	2	2	no	—
136	C₂×C₄×C₈₉	—	2	2	no	—
137	C₂×C₄×C₈₉	—	2	2	no	—
138	C₂×C₄×C₈₉	—	4	2	no	—
139	C₂×C₄×C₈₉	—	4	2	no	—
140	C₂×C₄×C₈₉	—	4	2	no	—
141	C₂×C₄×C₈₉	—	4	2	no	—
142	C₂×C₄×C₈₉	—	4	2	no	—
143	C₂×C₄×C₈₉	—	4	2	no	—
144	C₂×C₄×C₈₉	—	4	2	no	—
145	C₂×C₄×C₈₉	—	4	2	no	—
146	C₂×C₄×C₈₉	—	4	2	no	—
147	C₂×C₄×C₈₉	—	4	2	no	—
148	C₂×C₄×C₈₉	—	4	2	no	—
149	C₂×C₄×C₈₉	C₂²⋊Q₈	1	4	yes	—
150	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
151	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
154	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
156	C₈×C₈₉	—	2	2	no	—
157	C₈×C₈₉	—	2	2	no	—
158	C₈×C₈₉	—	4	1	no	—
159	C₈×C₈₉	—	4	1	no	—
160	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
161	C₂²×C₈₉	#532	1	1	yes	characteristic
162	C₂²×C₈₉	#535	1	1	yes	characteristic
163	C₂²×C₈₉	#577	1	2	yes	—
164	C₂²×C₈₉	#578	1	2	yes	—
165	C₄×C₈₉	—	2	4	no	—
166	C₄×C₈₉	—	4	4	no	—
167	C₄×C₈₉	—	4	1	no	—
168	C₄×C₈₉	—	4	1	no	—
169	C₄×C₈₉	—	4	2	no	—
170	C₄×C₈₉	—	8	2	no	—
171	C₄×C₈₉	—	8	2	no	—
172	C₄×C₈₉	—	8	2	no	—
173	C₄×C₈₉	#553	1	1	yes	characteristic
174	C₄×C₈₉	#553	1	1	yes	characteristic
175	C₄×C₈₉	#554	1	1	yes	characteristic
176	C₄×C₈₉	#554	1	1	yes	characteristic
177	#29202	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
178	C₂×C₈₉	#2566	1	1	yes	characteristic
179	C₂×C₈₉	#3082	1	2	yes	—
180	C₂×C₈₉	#3174	1	1	yes	characteristic
181	C₂×C₈₉	#3177	1	1	yes	characteristic
182	C₂×C₈₉	#3225	1	2	yes	—
183	#2647	C₂×C₈₉	1	1	yes	characteristic
184	#2327	C₂×C₈₉	1	1	yes	characteristic
185	#2068	C₂×C₈₉	1	2	yes	—
186	#2066	C₂×C₈₉	1	2	yes	—
187	#1960	C₂×C₈₉	1	1	yes	characteristic
188	#1956	C₂×C₈₉	1	1	yes	characteristic
189	#1865	C₂×C₈₉	1	2	yes	—
190	#1863	C₂×C₈₉	1	2	yes	—
191	#1846	C₂×C₈₉	1	1	yes	characteristic
192	#1836	C₂×C₈₉	1	1	yes	characteristic
193	#1768	C₂×C₈₉	1	1	yes	characteristic
194	C₈₉	#29202	1	1	yes	characteristic, direct product factor, Sylow
195	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
196	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
197	C₂×#107	C₂²×C₈₉	1	2	yes	—
198	C₂×#106	C₂²×C₈₉	1	2	yes	—
199	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
200	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
201	C₂×#103	C₂²×C₈₉	1	2	yes	—
202	C₂×#103	C₂²×C₈₉	1	2	yes	—
203	C₂×#103	C₂²×C₈₉	1	2	yes	—
204	C₂×#103	C₂²×C₈₉	1	2	yes	—
205	C₂×#97	C₂²×C₈₉	1	1	yes	characteristic
206	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
207	#400	—	2	2	no	—
208	#398	—	2	2	no	—
209	#391	C₂²×C₈₉	1	2	yes	—
210	#389	C₂²×C₈₉	1	2	yes	—
211	#389	C₂²×C₈₉	1	2	yes	—
212	#387	—	2	2	no	—
213	#387	—	2	2	no	—
214	#384	C₂²×C₈₉	1	1	yes	characteristic
215	#382	C₂²×C₈₉	1	1	yes	characteristic
216	#382	C₂²×C₈₉	1	1	yes	characteristic
217	#376	C₂²×C₈₉	1	1	yes	characteristic
218	#340	C₂²×C₈₉	1	2	yes	—
219	#340	C₂²×C₈₉	1	2	yes	—
220	#336	C₂²×C₈₉	1	1	yes	characteristic
221	#336	C₂²×C₈₉	1	1	yes	characteristic
222	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
223	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
224	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
225	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
226	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
227	C₄⋊Q₈	—	2	2	no	—
228	C₄⋊Q₈	Q₈×C₈₉	1	4	yes	—
229	C₄×Q₈	—	4	2	no	—
230	C₄×Q₈	—	4	2	no	—
231	C₂×C₄⋊C₄	—	2	2	no	—
232	C₂×C₄⋊C₄	—	2	2	no	—
233	C₂×C₄⋊C₄	—	2	2	no	—
234	C₂×C₄⋊C₄	—	2	2	no	—
235	C₂×C₄⋊C₄	—	2	2	no	—
236	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
237	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
238	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
239	C₂×C₄²	—	2	2	no	—
240	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
241	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
242	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
243	#107	—	4	2	no	—
244	#106	—	4	2	no	—
245	C₄⋊C₈	—	2	2	no	—
246	C₄⋊C₈	—	2	2	no	—
247	#103	—	2	4	no	—
248	#103	—	2	4	no	—
249	#103	—	4	2	no	—
250	#103	—	4	2	no	—
251	#97	—	2	2	no	—
252	C₄×C₈	—	2	2	no	—
253	#95	—	2	2	no	—
254	#95	—	2	2	no	—
255	#95	—	2	2	no	—
256	#95	—	2	2	no	—
257	#95	—	2	2	no	—
258	#95	—	2	2	no	—
259	#95	D₄×C₈₉	1	2	yes	—
260	#95	D₄×C₈₉	1	2	yes	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	C₂×Q₈×C₈₉	1	4	yes	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
272	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
274	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
275	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	#41×C₈₉	1	2	yes	—
287	C₂×C₈	#41×C₈₉	1	2	yes	—
288	C₄⋊C₄	—	2	8	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	4	4	no	—
296	C₄⋊C₄	—	8	2	no	—
297	C₄⋊C₄	—	8	2	no	—
298	C₄⋊C₄	—	8	2	no	—
299	C₄²	—	2	1	no	—
300	C₄²	—	2	1	no	—
301	C₄²	—	2	1	no	—
302	C₄²	—	2	1	no	—
303	C₄²	—	4	4	no	—
304	C₄²	C₂×Q₈×C₈₉	1	2	yes	—
305	C₄²	#41×C₈₉	1	2	yes	—
306	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
307	Q₈	—	2	4	no	—
308	Q₈	—	2	4	no	—
309	Q₈	—	4	2	no	—
310	Q₈	—	4	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	C₂²⋊Q₈×C₈₉	1	4	yes	—
333	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
334	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
335	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
336	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
337	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
338	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
339	C₈	—	2	2	no	—
340	C₈	—	2	2	no	—
341	C₈	—	4	1	no	—
342	C₈	—	4	1	no	—
343	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
344	C₂²	#532×C₈₉	1	1	yes	characteristic
345	C₂²	#535×C₈₉	1	1	yes	characteristic
346	C₂²	#577×C₈₉	1	2	yes	—
347	C₂²	#578×C₈₉	1	2	yes	—
348	C₄	—	2	4	no	—
349	C₄	—	4	4	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	1	no	—
352	C₄	—	4	2	no	—
353	C₄	—	8	2	no	—
354	C₄	—	8	2	no	—
355	C₄	—	8	2	no	—
356	C₄	#553×C₈₉	1	1	yes	characteristic
357	C₄	#553×C₈₉	1	1	yes	characteristic
358	C₄	#554×C₈₉	1	1	yes	characteristic
359	C₄	#554×C₈₉	1	1	yes	characteristic
360	C₂	C₈₉×#2566	1	1	yes	characteristic
361	C₂	C₈₉×#3082	1	2	yes	—
362	C₂	C₈₉×#3174	1	1	yes	characteristic
363	C₂	C₈₉×#3177	1	1	yes	characteristic
364	C₂	C₈₉×#3225	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ #376×C₈₉ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
177. #29202	C₈₉	direct (N × H)
194. C₈₉	#29202	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₈₉×#2647	2
C₈₉×#2327	2
C₈₉×#2068	2
C₈₉×#2066	2
C₈₉×#1960	2
C₈₉×#1956	2
C₈₉×#1865	2
C₈₉×#1863	2
C₈₉×#1846	2
C₈₉×#1836	2
C₈₉×#1768	2
#29202	89

Subgroups — C89×#29202 #36407050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#29202 #36407050